

A084386


Number of pairs of rabbits when there are 3 pairs per litter and offspring reach parenthood after 3 gestation periods.


8



1, 1, 1, 4, 7, 10, 22, 43, 73, 139, 268, 487, 904, 1708, 3169, 5881, 11005, 20512, 38155, 71170, 132706, 247171, 460681, 858799, 1600312, 2982355, 5558752, 10359688, 19306753, 35983009, 67062073, 124982332, 232931359, 434117578, 809064574
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OFFSET

0,4


COMMENTS

This comment covers an infinite family of growth sequences, where a(n) = a(n1) + k*a(nm). k is number of pairs per litter and m is periods until adulthood. G.f. = 1/(1xk*x^m). For example, A000930 has k=1 and m=3 while A006130 has k=3 and m=2.
The compositions of n in which each natural number is colored by one of p different colors are called pcolored compositions of n. For n>=3, 4*a(n3) equals the number of 4colored compositions of n with all parts >=3, such that no adjacent parts have the same color.  Milan Janjic, Nov 27 2011
a(n+2) equals the number of words of length n on alphabet {0,1,2,3}, having at least two zeros between every two successive nonzero letters.  Milan Janjic, Feb 07 2015
Number of compositions of n into one sort of part 1 and three sorts of part 3 (see the g.f.).  Joerg Arndt, Feb 07 2015


LINKS

Table of n, a(n) for n=0..34.
Merrill Jensen, Generating Functions
Index entries for linear recurrences with constant coefficients, signature (1,0,3).


FORMULA

a(n) = a(n1) + 3*a(n3). a(n) = A052900(n+3)/3.
G.f.: 1/(1x3*x^3).
a(n) = sum{k=0..floor(n/2), C(n2k, k)3^k}  Paul Barry, Nov 18 2003
G.f.: W(0)/2, where W(k) = 1 + 1/(1  x*(1 + 3*x^2)/(x*(1 + 3*x^2) + 1/W(k+1) )); (continued fraction).  Sergei N. Gladkovskii, Aug 28 2013


MAPLE

seq(add(binomial(n2*k, k)*3^k, k=0..floor(n/3)), n=0..34);  Zerinvary Lajos, Apr 03 2007


MATHEMATICA

a[0]=a[1]=a[2]=1; a[n_] := a[n]=a[n1]+3a[n3]; Table[a[n], {n, 0, 34}]
LinearRecurrence[{1, 0, 3}, {1, 1, 1}, 37] (* Robert G. Wilson v, Jul 12 2014 *)


PROG

(PARI) a(n)=([0, 1, 0; 0, 0, 1; 3, 0, 1]^n*[1; 1; 1])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016


CROSSREFS

Partial sums of A052900. Also A052900/3.
Cf. A000930, A006130, A001045.
Sequence in context: A227686 A161863 A102649 * A275176 A024726 A024948
Adjacent sequences: A084383 A084384 A084385 * A084387 A084388 A084389


KEYWORD

easy,nonn


AUTHOR

Merrill Jensen (mpjensen(AT)mninter.net), Jun 23 2003


EXTENSIONS

Edited by Dean Hickerson, Jun 24 2003


STATUS

approved



